Method for the higher order blind demodulation of a linear wave-shaped emitter

ABSTRACT

Process for the blind demodulation of a linear waveform source or transmitter in a system comprising one or more sources and an array of sensors and a propagation channel. The process comprises at least the following steps:  
     the symbol period T is determined and sampled to Te such that T=IT e  (I being an integer);  
     a spatio-temporal observation z(t), the mixed sources of which are symbol trains from the transmitter, is constructed from the observations x(kT e );  
     an ICA-type method is applied to the observation vector z(t) in order to estimate the L c  symbol trains {a m-i } that are associated with the channel vectors ĥ z,j =ĥ z (k j );  
     the L c  outputs (â m,j , ĥ z,j ) are arranged in the same order as the inputs (a m-i ,h z (i)) so as to obtain the propagation channel vectors ĥ z,j =ĥ z (k j ); and  
     the phase α imax  associated with the outputs is determined.

The object of the invention relates to a process for the blind demodulation of signals output by several transmitters and received by an array made up of at least one sensor.

For example, it applies to an array of antennas in an electromagnetic context.

The subject of the invention is in particular the demodulation of signals, that is to say the extraction of the symbols {a_(k)} transmitted by a linearly modulated transmitter.

FIG. 1 shows an antenna processing system comprising several transmitters E_(i) and an antenna processing system T comprising several antennas R_(i) receiving from radio sources at different angles of incidence. The angles of incidence of the sources or transmitters may be parameterized either in 1D with the azimuth θ_(m), or in 2D with the azimuth angle θ_(m) and the elevation angle Δ_(m).

FIG. 3 shows schematically a modulation/demodulation principle for the symbols {a_(k)} output by a transmitter. The signal propagates via a multipath channel. The transmitter outputs the symbol a_(k) at the instant k.T, where T is the symbol period. The demodulation consists in estimating and in detecting the symbols in order to obtain, at the output of the demodulator, the estimated symbols â_(k). In this figure, the train of symbols {a_(k)} is linearly filtered upon transmission by a transmission filter H, also called a wave-shaping filter h₀(t).

In the rest of the description, the expression “blind demodulation” is understood to mean techniques that basically use no information on the signal output, examples being a wave-shaping filter, a learning sequence, etc.

The last ten years have seen the development of SIMO (single-input, multiple-output) blind demodulation techniques called subspace techniques using 2nd-order statistics, as described in reference [7]. However, these algorithms have the drawback of not being robust to either underestimation or overestimation of the order of the propagation channel, resulting in temporal spreading dependent on the multipaths and on the wave-shaping filter. A linear prediction technique, described in reference [11], has been proposed for overcoming this problem, but this has the drawback of being less effective when the length of the channel is known. To improve the subspace techniques, the method described in [16] proposes a parametric technique, but unfortunately this requires knowledge of the wave-shaping filter.

In reference [13], the authors propose a technique based on covariance matching, but this has in particular the drawback of being very difficult to implement. An easier but suboptimal technique, described in reference [12], was therefore developed by minimizing a likelihood criterion and assuming the symbols to be Gaussian in character. This assumption is not verified for the widely used linear modulations such as PSK (Phase Shift Keying) or QAM (Quadrature Amplitude Modulation).

It is also known, in CMA (Constant Modulus Algorithm) methods, to use a spatio-temporal approach described for example in reference [6]. However, this family of methods has the drawback of being suitable only for one particular class of modulations, such as PSK, which are constant-modulus modulations. This method is iterative and therefore has the drawback of having to be correctly initialized. Finally, the CMA methods have the disadvantage of converging more slowly than the abovementioned subspace method. Moreover, reference [20] describes a subspace method making use of higher-order statistics for non-minimum-phase FIR (finite impulse response) channel identification.

The subject of the present invention is a process based in particular on blind source separation techniques known to those skilled in the art and described for example in references [4], [5], [15] and [19] assuming that the symbols transmitted are statistically independent. To do this, the process constructs a spatio-temporal observation whose mixed sources are symbol trains from the transmitter. Each symbol train is for example the same symbol train but shifted by an integral number of symbol periods T.

The invention relates to a process for the blind demodulation of a linear-waveform source or transmitter in a system comprising one or more sources and an array of sensors and a propagation channel, said process being characterized in that it comprises at least the following steps:

-   -   the symbol period T is determined and samples are taken at T_(e)         such that T=IT_(e) (I being an integer);     -   a spatio-temporal observation z(t), the mixed sources of which         are symbol trains from the transmitter, is constructed from the         observations x(kT_(e));     -   an ICA-type method is applied to the observation vector z(t) in         order to estimate the L_(c) symbol trains {a_(m-i)} that are         associated with the channel vectors ĥ_(z,j)=ĥ_(z)(k_(j));     -   the L_(c) outputs (â_(m,j), ĥ_(z,j)) are arranged in the same         order as the inputs (a_(m-i), h_(z)(i)) so as to obtain the         propagation channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)); and     -   the phase α_(imax) associated with the outputs is determined.

The process according to the invention offers in particular the following advantages:

-   -   it makes no assumption about the symbol constellations, unlike         the methods described in the prior art;     -   it requires no knowledge of the wave-shaping filter;     -   the modulus of the symbols is not assumed to be constant;     -   it is robust to channel length overestimation;     -   it can handle propagation channels with correlated paths; and     -   it is direct and simple to implement, with no correlated-path         crosscheck step.

Other features and advantages of the subject of the present invention will become more clearly apparent on reading the following description, given by way of illustration but implying no limitation, and on examining the appended figures, which show:

FIG. 1, an example of an architecture;

FIG. 2, the angles of incidence of the sources;

FIG. 3, the linear modulation and demodulation process for a symbol train;

FIG. 4, the diagram of a linear-modulation transmitter;

FIG. 5, a summary of the general principle employed in the invention;

FIG. 6, the representation of a constellation;

FIG. 7, a first example of the implementation of the method, in which the signal is received baseband;

FIG. 8, a second example in which the signal is received in baseband and the multipaths are decorrelated; and

FIG. 9, a third example in which the signal is received in baseband and the multipaths are groupwise decorrelated.

To explain the process according to the invention more clearly, the following description relates to a process for the higher-order blind demodulation of a linear-waveform transmitter in an array having a structure as described in FIG. 1 for example.

Before explaining the steps for implementing the process, the model of the signal used will be described.

Model of the Signal Output by a Source or Transmitter Linear Modulation

FIGS. 3 and 4 show the process for the linear modulation of a symbol train {a_(k)} at the rate T by a wave-shaping filter h₀(t).

The comb of symbols c(t) is firstly filtered by the wave-shaping filter h₀(t) and then transposed to the carrier frequency f₀. The NRZ filter, which is a time window of length T, very often defined by h₀(t)=Π_(T)(t−T/2), is one particular nonlimiting example of a transmission filter. In radio communication, it is also possible to use a Nyquist filter, the Fourier transform of which, h₀(f)≈Π_(B)(f−B/2), approaches a band window B when the roll-off is zero, therefore h₀(f)=Π_(B)(f−B/2) (the roll-off defines the slope of the filter away from the band B).

The modulates signal s₀(t), output by the transmitter, may be expressed at time t_(k)=kT_(e) (T_(e) being the sampling period) as a function of the comb of symbols c(t): $\begin{matrix} {{s_{0}\left( {kT}_{e} \right)} = {\sum\limits_{i}^{\quad}{{h_{0}\left( {iT}_{e} \right)}{c\left( {\left( {k - i} \right)T_{e}} \right)}}}} & (1) \end{matrix}$

Let the symbol time T be equal to an integer number of times the sampling period, i.e. T=IT_(e), and let k=mI+j where 0≦j<I. Since c(t)=Σ_(r)a_(r)δ(t−rIT_(e)), in other words c(t)=a_(u) for t=uIT_(e) and c(t)=0 for t≠uIT_(e), the only values for i for which c((k−i)T_(e)) is non zero satisfy the equation k−i=uI, that is to say such that i=mI+j−ul=nI+j, where n=m−u. Finally, equation (1) becomes: $\begin{matrix} {{s_{0}\left( {{mIT}_{e} + {jT}_{e}} \right)} = {{\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{0}\left( {{nIT}_{e} + {jT}_{e}} \right)}a_{m - n}\quad{for}\quad 0}} \leq j < I}} & (2) \end{matrix}$

The parameter L₀ is the half-length of the transmission filter which is spread over a duration of (2L₀+1)IT_(e). In the particular case of an NRZ transmission filter, L₀=0. As regards the transmitted signal s(t), this satisfies the equation s(t)=s₀(t)exp(j2Πf₀t) as it is equal to the signal s₀(t) transposed to the frequency f₀. Under these conditions, the expression s(mIT_(e)+jT_(e)) is, from (2): $\begin{matrix} {\begin{matrix} {{s\left( {{mIT}_{e} + {jT}_{e}} \right)} = {\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{0}\left( {{nIT}_{e} + {jT}_{e}} \right)}{\exp\left( {{j2\pi}\quad{f_{0}\left( {{nI} + j} \right)}T_{\underset{\_}{e}}} \right)}}}} \\ {a_{m - n}{\exp\left( {{j2\pi}\quad{f_{0}\left( {m - n} \right)}{IT}_{e}} \right)}} \\ {= {{\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{F\quad 0}\left( {{nIT}_{e} + {jT}_{e}} \right)}b_{m - n}\quad{such}\quad{that}\quad 0}} \leq j < I}} \end{matrix}{{where}\quad\begin{matrix} {{h_{F\quad 0}\left( {iT}_{e} \right)} = {{h\left( {iT}_{e} \right)}{\exp\left( {{j2}\quad\pi\quad f_{0}{iT}_{e}} \right)}\quad{and}}} \\ {b_{i} = {a_{i}{\exp\left( {{j2\pi}\quad f_{0}{iIT}_{e}} \right)}}} \end{matrix}}} & (3) \end{matrix}$

Reception of the Signals by the Sensors

The transmitted signal s(t) (FIG. 3) passes through a propagation channel before being received on an array made up of N antennas. The propagation channel may be modelled by P multipaths of angle of incidence θ_(p), delay τ_(p) and amplitude ρ_(p) (1≦p≦P). At the output of the antennas is the vector x(t), which corresponds to the sum of a linear mixture of P multipaths and noise, assumed to be white and Gaussian. This vector of dimension N×1 is given by the following expression: $\begin{matrix} \begin{matrix} {{x(t)} = {{\sum\limits_{p = 1}^{P}{\rho_{p}{a\left( \theta_{p} \right)}{s\left( {t - \tau_{p}} \right)}}} + {b(t)}}} \\ {\quad{= {{A\quad{s(t)}} + {b(t)}}}} \end{matrix} & (4) \end{matrix}$

where ρ_(p) is the amplitude of the pth path, b(t) is the noise vector, assumed to be Gaussian, a(θ) is the response of the array of sensors to a source with angle of incidence θ, and A=[a(θ₁) . . . a(θ_(p))] and s(t)=[s(t−τ₁) . . . s(t−τ_(p))]^(T). Noting that τ_(p)=r_(p)T+Δτ_(p) (where 0≦Δτ_(p)<T=IT_(e)and r_(p) is an integer), and inserting equation (3) into equation (4), the vector received by the antennas is given by: $\begin{matrix} {{x\left( {{mIT}_{e} + {jT}_{e}} \right)} = {{\sum\limits_{p = 1}^{P}{\sum\limits_{\quad{n = {- L_{0}}}}^{L_{0}}{\rho_{p}{a\left( \theta_{p} \right)}{h_{F\quad 0}\left( {{nIT}_{e} + {jT}_{e} - {\Delta\tau}_{p}} \right)}{b_{m - n - r}}_{p}}}} + {b\left( {{mIT}_{e\quad} + {jT}_{e}} \right)}}} & (5) \end{matrix}$

By making the change of variable according to u_(p)=n+r_(p), the vector received by the antennas is given by: $\begin{matrix} {{x\left( {{mIT}_{e} + {jT}_{e}} \right)} = {{\sum\limits_{p = 1}^{P}{\sum\limits_{{u_{p} = {r_{p} - L_{0}}}\quad}^{r_{p} + L_{0}}\quad{\rho_{p}{a\left( \theta_{p} \right)}{h_{F\quad 0}\left( {{\left( {u_{p} - r_{p}} \right){IT}_{e}} + {jT}_{e\quad} - {\Delta\tau}_{p}} \right)}b_{m - u_{p}}}}} + {b\left( {{mIT}_{e} + {jT}_{e}} \right)}}} & (6) \end{matrix}$

Notating r_(min)=min{r_(p)} and r_(max)=max{r_(p)}, equation (6) may be rewritten as follows: $\begin{matrix} {{x\left( {{mIT}_{e} + {jT}_{e}} \right)} = {{\sum\limits_{p = 1}^{P}{\sum\limits_{u = {r_{\min} - L_{0}}}^{r_{\min} + L_{0}}{\rho_{p}{a\left( \theta_{p} \right)}{h_{F\quad 0}\left( {{\left( {u - r_{p}} \right){IT}_{e}} + {jT}_{e\quad} - {\Delta\tau}_{p}} \right)}{{Ind}_{\lbrack{{{rp} - {L\quad 0}},{{rp} + {L\quad 0}}}\rbrack}(u)}b_{m - u}}}} + {b\left( {{mIT}_{e} + {jT}_{e}} \right)}}} & (7) \end{matrix}$

where Ind_([r,q])(u) is the usual indicatrix function (Ind_([r,q])(u)=1 for r≦u≦p and Ind_([r,q])(u)=0 otherwise) defined over the set of integers relating to the value in the binary set {0,1}, characterized by Ind_([r,q])(u)=1 if u belongs to the interval [r,q] and Ind_([r,q])(u)=0 otherwise. Thus, denoting the channel vector by v(t): $\begin{matrix} {{v\left( {{uIT}_{e} + {jT}_{e}} \right)} = {\sum\limits_{p = 1}^{P}{\rho_{p}{a\left( \theta_{p} \right)}{h_{F\quad 0}\left( {{\left( {u - r_{p}} \right){IT}_{e}} + {jT}_{e\quad} - {\Delta\tau}_{p}} \right)}{{Ind}_{\lbrack{{{rp} - {L\quad 0}},{{rp} + {L\quad 0}}}\rbrack}(u)}}}} & (8) \end{matrix}$

where t=uIT_(e)+jT_(e) and equation (5) becomes: $\begin{matrix} {{{x\left( {{mIT}_{e} + {jT}_{e}} \right)}{\sum\limits_{u = {r_{\min} - L_{0}}}^{r_{\min} + L_{0}}{{v\left( {{uIT}_{e} + {jT}_{e}} \right)}b_{m - u}}}} + {b\left( {{mIT}_{e} + {jT}_{e}} \right)}} & (9) \end{matrix}$

Inter-Symbol Interference

The observation vector x(t) coming from the antenna array at t=mIT_(e)+jT_(e) involves, from equation (9), the symbol b_(m), but also the symbols b_(m-u) where u is a relative integer lying within the [r_(min)−L₀, r_(max)+L₀] interval, which phenomenon is more widely known as ISI (inter-symbol interference). Let L_(c) be the number of symbols participating in the ISI and let the interval of values taken by the latter be limited. From equation (9), if the intersection of the intervals [r_(p)−L₀, r_(p)+L₀] is non-empty, then L_(c)=|r_(max)−r_(min)|+2L₀+1. Consequently, when r_(max)=r_(min), that is to say when all the multipaths are correlated, the lower bound of L_(c) is reached and is given by L_(c)=2L₀+1. This case is also mathematically expressed by ${{{\max\limits_{p}\left\{ \tau_{p} \right\}} - {\min\limits_{p}\left\{ \tau_{p} \right\}}}} < {T.}$ On the other hand, if the intersection of said intervals is empty and if all the intervals [r_(p)−L₀, r_(p)+L₀] are disjoint, then L_(c)=P×(2L₀+1), which constitutes an upper bound for the set of values that can be taken by L_(c). The latter situation corresponds specifically to the case of multipaths that are all pairwise decorrelated. This may also be mathematically expressed as ∀i≠j, |r_(i)−r_(j)|>2L₀, this condition being obtained whenever |τ_(i)−τ_(j)|>(2L₀+1)T. To summarize, the quantity L_(c) is in general bounded as follows: 2L ₀+1≦L _(c) ≦P×(2L ₀+1)   (10)

The equation expressing the vector received by the sensors can then be rewritten in the following manner, but this time only the L_(c) symbols b_(m-u) of interest appear: $\begin{matrix} {{X\left( {{mIT}_{e} + {jT}_{e}} \right)} = {{\sum\limits_{l = 1}^{L_{0}}{{h\left( {{{n(I)}{IT}_{e}} + {jT}_{e}} \right)}b_{m - {n{(I)}}}}} + {b\left( {{mIT}_{e} + {jT}_{e}} \right)}}} & (11) \end{matrix}$

where ∀1≦I≦L_(c) and r_(min)−L₀≦n(I)≦r_(min)+L₀ and where: $\begin{matrix} {{h(t)} = {\sum\limits_{p = 1}^{P}{\rho_{p}{a\left( \theta_{p} \right)}{h_{F0}\left( {t - \tau_{p}} \right)}}}} & (12) \end{matrix}$

ICA Techniques

The process uses ICA techniques based on the following model, given by way of entirely nonlimiting illustration: $\begin{matrix} \begin{matrix} {u_{k} = {{\sum\limits_{i = 1}^{\quad L}{g_{i}s_{ik}}} + n_{k}}} \\ {= {{Gs}_{k} + n_{k}}} \end{matrix} & (13) \end{matrix}$

where u_(k) is a vector of dimension M×1 received at time k, s_(ik) is the ith component of the signal s_(k) at time k, n is the noise vector and G=[g₁ . . . g_(L)]. The objective of the ICA methods is to extract the I=L components s_(ik) and to identify their signatures g_(i) (the vectorial response of source i through the observation u_(k)) on the basis of the observation u_(k). The number I=L of components must not exceed the dimension M of the observation vector. The methods of references [4], [5] and [15] use 2nd- and 4th-order statistics of the observations u_(k). The first step uses 2nd-order statistics for the observations u_(k) (these observations may be functions of the signals received by the sensors) in order to obtain a new observation z_(k), such that: $\begin{matrix} \begin{matrix} {z_{k} = {W_{1}u_{k}}} \\ {= {{\sum\limits_{i = 1}^{\quad L}{{\overset{˘}{g}}_{i}s_{ik}}} + {\overset{\sim}{n}}_{k}}} \\ {= {{\overset{˘}{G}s_{k}} + {\overset{\sim}{n}}_{k}}} \end{matrix} & (14) \end{matrix}$

where the signatures {hacek over (g)}_(i) (1≦i≦L) are orthogonal, {hacek over (G)}=[{hacek over (g)}₁ . . . {hacek over (g)}_(L)] and s_(k)=[s_(1k) . . . s_(Lk)]^(T). The second step consists in identifying the orthogonal base of the {hacek over (G)} values using 4th order statistics of the whitened observations z_(k). In this way, the signals s_(k) may be extracted by effecting: gk=GZk=G W. Uk (15)

where Ŝ_(k) is the estimate of the signals s_(k) and where # is the pseudo-inversion operator defined by {hacek over (G)}^(#)=({hacek over (G)}^(H){hacek over (G)})⁻¹{hacek over (G)}^(H).

The ICAR method [19] uses only 4th-order statistics to identify the matrix G=[g₁ . . . g_(k)] of signatures.

To summarize, the idea employed in the process according to the invention is to construct a spatio-temporal observation, the mixed sources of which are symbol trains from the transmitter. Each symbol train is for example the same symbol train but shifted by an integral number of symbol periods T.

Several ways of implementing the method will be described below, some of which are explained by way of non-limiting illustration.

First Way of Implementing the Process.

FIG. 7 shows a first illustrative way of implementing the process in which the signal is received in baseband.

The method comprises a step I.1 of determining the symbol time T_(e), for example by applying a cyclic detection algorithm, such as that described for example in [1] [10].

The next step I.2 consists in interpolating the observations x(t) with I samples per symbol, such that T=IT_(e).

Under these conditions where f₀=0 and b_(k)=a_(k), equation (11) for the vector becomes: $\begin{matrix} {{x\left( {{mIT}_{e} + {jT}_{e}} \right)} = {{{\sum\limits_{l = 1}^{L_{0}}{{h\left( {{{n(I)}{IT}_{e}} + {jT}_{e}} \right)}a_{m - {n{(I)}}}}} + {{b\left( {{mIT}_{e} + {jT}_{e}} \right)}\quad{for}\quad 0}} \leq j < I}} & (16) \end{matrix}$

Since equation (16) is valid for 0≦j<I, the method constructs the next spatio-temporal observation (step I3) from the observations x(kT_(e)): $\begin{matrix} {\begin{matrix} {{z\left( {mIT}_{e} \right)} = \begin{bmatrix} \begin{matrix} \begin{matrix} {x\left( {mIT}_{e} \right)} \\ {x\left( {{mIT}_{e} + T_{e}} \right)} \end{matrix} \\ \vdots \end{matrix} \\ {x\left( {{mIT}_{e} + {\left( {I - 1} \right)T_{e}}} \right)} \end{bmatrix}} \\ {= {{\sum\limits_{l = 1}^{L_{c}}{{h_{z}\left( {n(I)} \right)}a_{m - {n{(I)}}}}} + {{b_{z}\left( {mIT}_{e} \right)}\quad{where}\quad{h_{z}(n)}}}} \\ {= \begin{bmatrix} \begin{matrix} \begin{matrix} h_{n,0} \\ h_{n,1} \end{matrix} \\ \vdots \end{matrix} \\ h_{n,{I - 1}} \end{bmatrix}} \end{matrix}\begin{matrix} {{{with}\quad h_{n,j}} = {{h\left( {{nIT}_{e} + {jT}_{e}} \right)}\quad{and}}} \\ {{b_{z}\left( {mIT}_{e} \right)} = {\left\lbrack {{b\left( {mIT}_{e} \right)}^{T}\ldots\quad{b\left( {{mIT}_{e} + {\left( {I - 1} \right)T_{e}}} \right)}^{T}} \right\rbrack^{T}.}} \end{matrix}} & (17) \end{matrix}$

Since it is known that x(t) has the dimensions N×1, the vector z(t) has the dimensions NI×1.

h(k) is a vector whose nth component is the kth component of the filter that linearly filters the symbol train {a_(m)} on the nth sensor. The filter for the vector coefficient h(k) depends both on the wave-shaping filter and on the propagation channel.

To extract the L_(c) symbol trains {a_(m-i)} of interest (the number of symbols participating in the ISI), the method samples the received signal with I=(2L₀+1), assuming that P≦N.

Since it is known that the NRZ filter satisfies 2L₀+1=1 and the Nyquist filter 2L₀+1=3 for a roll-off of 0.25, the symbol trains may be extracted for these two wave-shaping filters when P≦NI and 3P≦NI, respectively.

Having determined the observation vector z(t), the process applies an ICA-type method to estimate the L_(c) symbol trains {a_(m-i)} associated with the channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)).

The jth output of the ICA method gives the symbol train {â_(m,j)} associated with the channel vector ĥ_(z,j). The estimated symbol trains {â_(m,j)} arrive in a different order from that of the {a_(m-i)} trains satisfying: â _(m,j) =ρexp(jα _(i))a _(m-i) and ĥ _(z,j) =ĥ _(z)(i)   (18)

The symbol trains {â_(m,j)} are estimated with the same amplitude because the symbol trains {a_(m-i)} all have the same power, satisfying the equation: E[|a _(m-n(1))|² ]=. . . =E[|a _(m-n(Lc))|²].

The next step I.4 of the process has the objective of ordering the L_(c) outputs (â_(m,j,)ĥ_(z,j)) in the same order as the inputs (a_(m-i), h_(z)(i)) so as to obtain the channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)). To do this, the process intercorrelates pairwise the outputs â_(m,i) and â_(m,j), calculating the following criterion c_(i,j)(k): $\begin{matrix} {{c_{i,j}(k)} = \frac{E\left\lbrack {{\hat{a}}_{m,i}{\hat{a}}_{{m - k},j}^{*}} \right\rbrack}{\sqrt{{E\left\lbrack {{\hat{a}}_{m,i}{\hat{a}}_{m,i}^{*}} \right\rbrack}{E\left\lbrack {{\hat{a}}_{{m - k},j}{\hat{a}}_{{m - k},j}^{*}} \right\rbrack}}}} & (19) \end{matrix}$

When the function |c_(i,j)(k)| is a maximum in k=k_(max), the ith and jth outputs satisfy the equation: â_(m,i)=â_(m-k max,j). The algorithm for classifying the outputs â_(m,n(1)) . . . â_(m,n(Lc)) is for example composed of the following steps:

Step A.1: Determination of the output â_(m,imax) associated with the channel vector of higher-modulus ĥ_(z,jmax).

Step A.2: For all the outputs â_(m-k,j) where j≠i_(max), determination of the indices k=k_(j) maximizing the |c_(imax,j(k))| criterion. From this is deduced, for each j, that â_(m,imax)=â_(m-kj,j). Since it is known that c_(imax,j)(k_(j))=exp(jα_(imax)-jα_(j)) the jth output is reset to the same phase as the iith i_(max) output by taking â_(m-kj)=c_(imax,j)(k_(j))â_(m,j). The channel vectors are also reset in terms of phase by taking: {circumflex over (ĥ)}_(z)(k_(j))=ĥ_(z,j)c_(imax,j)(k_(j))*.

Step A.3: This step reorders the outputs â_(m-kj) and the channel vectors {circumflex over (ĥ)}_(z,j)=ĥ_(z)(k_(j)) in the increasing order of the K_(j), since it is known that â_(m)=â_(m,imax) and that {circumflex over (ĥ)}_(z)(0)=ĥ_(z,imax).

After these three steps, the symbol trains {â_(m-k)} associated with the channel vectors ĥ_(z)(k_(j)) are obtained. Since it is known that the estimated symbols satisfy the equation â_(m-k)=exp(jα_(imax))a_(m-k), the last step of the process consists in estimating this phase α_(imax). To do this, the constellation of symbols a_(k) is firstly identified among a database consisting of the set of possible constellations. This database consists of known constellations such as nPSK, n-QAM. Each time that a new constellation is detected or becomes known, this is added to the database.

FIG. 6 shows an example of an 8-QAM constellation when α_(imax)=0 and α_(imax)≠0. In this implementation example, the process then includes the following steps:

The next step I.5 consists in determining the output phase associated with the channel vector of higher modulus. To identify the constellation and determine the phase, the process performs, for example, the following steps:

Step I.5=Steps B.1, B.2 and B.3

Step°B.1: Estimation of the positions of the states of the constellation (red points in the figure) by seeking the maximum of the 2D histogramme of the points M_(k)=(real(â_(k)), imag(â_(k))). For a constellation consisting of M states, M pairs (û_(m),{circumflex over (v)}_(m)) for 1<m≦M are obtained.

Step°B.2 : Determination of the type of constellation by comparing the position of the states (û_(m),{circumflex over (v)}_(m)) of the constellation of {â_(k)} symbols with a database comprising the set of possible constellations. The closest constellation is made up of the states (u_(m),v_(m)) for 1≦m≦M.

Step B.3: Determination of the phase α_(imax) by minimizing in the sense of the least squares the following system of equations: û _(m)=cos(α_(imax))u _(m)−sin(α_(imax))v _(m and) {circumflex over (v)} _(m)=sin(α_(imax))u _(m)+cos(α_(imax))v _(m) for 1≦m≦M.

The process may include a step of estimating the propagation channel parameters of angle θ_(p) and delay τ_(p), of equation (8) by the algorithm proposed in [8]. The step consists in extracting firstly the vectors h(nIT_(e)+jT_(e)) for 0≦j<I from the channel vectors ĥ_(z)(n_(j)) defined in equation (17). Followed by construction of the matrix H=[h(n(1)IT_(e)) . . . h(n(L_(c))IT_(e))] from equation (11) with the h(nIT_(e)+jT_(e)) values in order to apply the parametric estimation method [8] for the multipaths: (θ_(p), τ_(p)) 1≦p≦p.

Second Way of Implementing the Process

FIGS. 8 and 9 show schematically another way of implementing the process, which may include two variants corresponding to the decorrelated multipath case and to the groupwise correlated multipath case, respectively.

Decorrelated Multipath Case.

The signal is received in baseband with {b_(k)}={a_(k)}.

The multipaths, the delays of which satisfy the relationship |τ_(j)-τ_(i)|>(2L₀+1)T, have the advantage of being decorrelated with one another, satisfying the equation: E[s(t−τ_(i))s(t−τ_(j))*]=0. By examining equation (4), it may therefore be seen that it is sufficient to apply an ICA type method when P≦N to the observation x(t) in order to obtain the signals s(t−τ_(p)) for each of the multibars. After estimating the signals for the various multipaths, the process determines their powers in order to keep the signal s(t−τ_(pmax)) of the multipath of higher amplitude ρ_(pmax). This main path is determined using the fact that the outputs of the ICA methods asymptotically satisfy: $\begin{matrix} \begin{matrix} {{x(t)} = {\sum\limits_{p = 1}^{P}{\rho_{p}{a\left( \theta_{p} \right)}{s\left( {t - \tau_{p}} \right)}}}} \\ {= {\sum\limits_{p = 1}^{P}{{\hat{a}}_{i}{{\hat{s}}_{i}(t)}\quad{with}}}} \\ {{{{\hat{s}}_{i}(t)} = {\frac{s\left( {t - \tau_{p}} \right)}{\sqrt{\gamma_{p}}}\quad{and}}}\quad} \\ {{\hat{a}}_{i} = {\sqrt{\gamma_{p}}\rho_{p}{a\left( \theta_{p} \right)}}} \end{matrix} & (20) \end{matrix}$

where γ_(p)=ρ_(p) ²E[|s(t−τ_(p))|²]. Since the vectors a(θ_(p)) are normed, satisfying the equation a(θ_(p))^(H)a(θ_(p))=N, the path of maximum amplitude will be associated with the i_(max) ^(th) output where α_(imax)=â_(imax) ^(H)â_(imax) is a maximum. From equation (3), the output ŝ_(imax)(t)=s(t−τ_(pmax)) satisfies the equation: $\begin{matrix} {{{\hat{s}}_{i\quad\max}\left( {{mIT}_{e} + {jT}_{e}} \right)} = {\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{F\quad 0}\left( {{nIT}_{e} + {jT}_{e} - \tau_{p\quad\max}} \right)}a_{m - n}}}} & (21) \end{matrix}$

such that 0≦j<I

and it is possible to constitute the following observation vector: $\begin{matrix} \begin{matrix} {{z\left( {mIT}_{e} \right)} = \begin{bmatrix} \begin{matrix} \begin{matrix} {{\hat{s}}_{i\quad\max}\left( {mIT}_{e} \right)} \\ {{\hat{s}}_{i\quad\max}\left( {{mIT}_{e} + T_{e}} \right)} \end{matrix} \\ \vdots \end{matrix} \\ {{\hat{s}}_{i\quad\max}\left( {{mIT}_{e} + {\left( {I - 1} \right)T_{e}}} \right)} \end{bmatrix}} \\ {= {\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{z}(n)}a_{m - n}}}} \\ {{{where}\quad{h_{z}(n)}} = {\begin{bmatrix} \begin{matrix} \begin{matrix} h_{n,0} \\ h_{n,1} \end{matrix} \\ \vdots \end{matrix} \\ h_{n,{I - 1}} \end{bmatrix}\quad{and}}} \end{matrix} & (22) \end{matrix}$

where h_(n,j)=h_(F0)(nIT_(e)+jT_(e)−τ_(pmax)). According to the model of equation (22), it is sufficient to apply an ICA method to the observation z(mIT_(e)) in order to estimate the 2L₀+1 symbol trains {a_(m-n)} with −L₀≦n≦L₀. To extract the angles of incidence θ_(p) of the propagation channel, it is sufficient from equation (20) to find, for each signature â_(i) (1≦i≦P), the maximum of criterion c(θ)=|a(θ)^(H)â_(i)|². To extract the delays τ_(i)-τ₁ of the propagation channel, it is sufficient from equation (20) to find, for each signal ŝ_(i)(t) (1≦i≦P), the maximum of the c(τ)=|ŝ_(i)(t−τ)ŝ_(l)(t)*|² criterion.

To summarize, this variant comprises, for example, the following steps:

Step II.a.1: Determination of the symbol period T, applying a cyclic detection algorithm as in [1] [10].

Step II.a.2: Sampling of the observations x(t) with I samples per symbol such that T=IT_(e).

Step II.a.3: Application of an ICA method to the observations x(t) in order to obtain ŝ_(i)(t) and â_(i) for 1≦i≦P.

Step II.a.4: Determination of the output i=imax where α_(i)=â_(i) ^(H)â_(i) is its maximum.

Step II.a.5: Formation of the observation vector z(t) of equation (22) from the signal ŝ_(imax)(t).

Step II.a.6: Application of an ICA method for estimating the symbol trains {a_(m-n)} where −L₀≦n≦L₀. From the symbol trains is chosen that one which is associated with the higher-modulus vector h_(z)(n), namely {â_(m)}.

Step II.a.7: Determination of the phase α_(imax) of the output associated with the higher-modulus vector h_(z)(n) applying steps B.1, B.2 et B.3.

Step II.a.8: Phase-resetting of the symbol train {â_(m)} by taking {circumflex over (â)}_(m)=â_(m)exp(−jα_(imax)). The symbol train {{circumflex over (â)}_(m)} constitutes the output of the demodulator of this subprocess.

Step II.a.9: Estimation of the propagation channel parameters, namely angle θ_(p) and delay τ_(p), by maximizing, for 1≦i≦P the |a(θ)^(H)â_(i)|² and |ŝ_(i)(t−τ)ŝ_(l)(t)*|² criteria for the angles and delays respectively.

General Case for any or Groupwise-Correlated Multipaths

In this variant, the diagram for which is given in FIG. 9, the process considers that some of the multipaths are correlated. Considering that the transmitter is received according to Q groups of correlated multipaths, the signal vector received by the sensors becomes, from equation (4): $\begin{matrix} \begin{matrix} {{x(t)} = {{\sum\limits_{q = 1}^{Q}\quad{\sum\limits_{p = 1}^{P_{q}}{\rho_{p,q}{a\left( \theta_{p,q} \right)}{s\left( {t - \tau_{p,q}} \right)}}}} + {b(t)}}} \\ {= {{\sum\limits_{q = 1}^{Q}{A_{q}\Omega_{q}{s\left( {t,{\underset{\_}{\tau}}_{q}} \right)}}} + {b(t)}}} \end{matrix} & (23) \end{matrix}$

Where A_(q)=[a(θ_(1,q)) . . . a(θ_(Pq,q))], Ω_(q)=diag([ρ_(1,q) . . . ρ_(Pq,q)]) and s(t, τ _(q))=[s(t−τ_(1,q)) . . . s(t−τ_(Pq,q))]^(T) with τ _(q)=[τ_(1,q) . . . τ_(Pq,q)]^(T). The following signals and signatures are estimated as output of the separator by applying an ICA method: $\begin{matrix} \begin{matrix} {\hat{A} = \left\lbrack {{\hat{a}}_{1}\quad\ldots\quad{\hat{a}}_{{PQ},Q}} \right\rbrack} \\ {= {\left\lbrack {A_{1}U_{1}\quad\ldots\quad A_{Q}U_{Q}} \right\rbrack\Pi\quad{and}}} \\ {{\hat{s}(t)} = {\Pi\begin{bmatrix} {V_{1\quad}{s\left( {t,{\underset{\_}{\tau}}_{1}} \right)}} \\ \vdots \\ {V_{Q}{s\left( {t,{\underset{\_}{\tau}}_{Q}} \right)}} \end{bmatrix}}} \\ {= \begin{bmatrix} {{\hat{s}}_{1}(t)} \\ \vdots \\ {{\hat{s}}_{P_{Q}x_{Q}}(t)} \end{bmatrix}} \end{matrix} & (24) \end{matrix}$

where Π is a permutation matrix, U_(q)V_(q)=Ω_(q) and V_(q)E[s(t,τ_(q))s(t,τ_(q))^(H)]V_(q) ^(H)=I_(Pq). Thus, the paths decorrelated such that E[s(t−τ_(p,q))s(t−τ_(p′,q′))*]=0 are received on different channels ŝ_(i)(t) and ŝ_(j)(t). The correlated paths where E[s(t−τ_(p,q))s(t−τ_(p′,q′))*]≠0 are mixed in the same channel ŝ_(i)(t) and are present on P_(Q) at the same time. In the 1st step of this subprocess, we use this result to identify the Q group of correlated multipaths. Taking the outputs i and j of the separator, the two following hypotheses may be tested: $\begin{matrix} {H_{0}\text{:}\left\{ {\begin{matrix} {{{\hat{s}}_{i}(t)} = {b_{i}(t)}} \\ {{{\hat{s}}_{j}(t)} = {b_{j}(t)}} \end{matrix}\quad{and}\quad H_{1}\text{:}\quad\left\{ \begin{matrix} {{{\hat{s}}_{i}(t)} = {{\alpha_{i}{s\left( {t - \tau_{p}} \right)}} + {b_{i}(t)}}} \\ {{{\hat{s}}_{j}(t)} = {{\alpha_{j}{s\left( {t - \tau_{p}} \right)}} + {b_{j}(t)}}} \end{matrix} \right.} \right.} & (25) \end{matrix}$

where E[b_(i)(t) b_(j)(t−τ)*]=0 whatever the value of τ. Thus for the H₀ hypothesis, no multipaths exist common to the two output i and j, and for the H₁ hypothesis there is at least one of them. The test consists in determining whether the outputs ŝ_(i)(t) and ŝ_(j)(t−τ) are correlated for at least one of the τ values satisfying |τ|<τ_(max). To do this, the Gardner test [3] is applied, which compares the following likelihood ratio with a threshold: $\begin{matrix} \begin{matrix} {{V_{ij}(\tau)} = {{- 2}K\quad{\ln\left( {1 - \frac{{{{\hat{r}}_{ij}(\tau)}}^{2}}{{{\hat{r}}_{ii}(0)}{{\hat{r}}_{jj}(0)}}} \right)}\quad{with}}} \\ {{{\hat{r}}_{ij}(\tau)} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{{{\hat{s}}_{i}(t)}{{\hat{s}}_{j}\left( {t - \tau} \right)}^{*}\quad{or}}}}} \\ \left. {{V_{ij}(\tau)} < \eta}\Rightarrow{{hypothesis}\quad H_{0}} \right. \\ \left. {{{And}\quad{V_{ij}(\tau)}} \geq \eta}\Rightarrow{{hypothesis}\quad H_{1}} \right. \end{matrix} & (26) \end{matrix}$

The threshold η is determined in [3] in relation to a chi square law with 2 degrees of freedom. The output associated with the 1st output are firstly sought by starting the test by 2<j<P_(Q)×Q and i=1. Next, removed from the list of outputs are all those associated with the 1st which will constitute the 1st group with q=1. The same series of tests is then restarted with the other outputs not correlated with the 1st output in order to constitute the 2nd group. This operation will be carried as far as the last group where, in the end, no output channel will remain. After the sorting, what will finally be obtained are: Â _(q) =A _(q) U _(q) and ŝ _(q)(t)=V _(q) s(t,T _(q)) for (1≦q≦Q)   (27)

The angles of incidence θ_(p,q) are determined from the Â_(q) values for (1≦q≦Q) applying the MUSIC [1] algorithm to the Â_(q)Â_(q) ^(H) matrix. The matrices A_(q) are deduced from these goniometry values. Since is it known that x_(q)(t)=Â_(q)ŝ_(q)(t)=A_(q)Ω_(q)s(t, τ _(q)), s(t, τ _(q)) is deduced therefrom to within a diagonal matrix by taking ŝ(t, τ _(q))=A_(q) ^(#)X_(q)(t). Since the elements of ŝ(t, τ _(q)) are composed of the signals ŝ(t−τ_(p,q)), the delays τ_(pq)−τ_(1,1) are determined by maximizing the c(τ)=|ŝ_(q,p)(t−τ)ŝ_(1,1)(t)*|² criteria where ŝ_(q,p)(t) is the p^(th) component of ŝ(t, τ _(q)).

Since it is known that E[ŝ_(q)(t)ŝ_(q)(t)^(H)]=I_(Pq), that A_(q) ^(H)A_(q)=N I_(Pq) and that Â_(q)ŝ_(q)(t)=A_(q)Ω_(q)s(t, τ _(q)), it is deduced therefrom that the group of multipaths associated with the largest amplitudes Ω_(q) maximizes the following criterion: cri(q)=trace(Â_(q) ^(H)Â_(q)). From this is deduced the best output associated with Â_(qmax) and ŝ_(qmax)(t). Since from equation (3) the vector s(t, τ _(qmax)) satisfies equation: $\begin{matrix} \begin{matrix} {{s\left( {{mlT}_{e} + {jT}_{e,{\underset{\_}{T}}_{q\quad\max}}} \right)} = \begin{bmatrix} {s\left( {{mlT}_{e} + {jT}_{e} - \tau_{q\quad\max\quad 1}} \right)} \\ \vdots \\ {s\left( {{mlT}_{e} + {jT}_{e} - \tau_{q\quad\max\quad P_{q\quad\max}}} \right)} \end{bmatrix}} \\ {= {\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{F\quad 0}\left( {{nlT}_{e} + {jT}_{e,{\underset{\_}{T}q\quad\max}}} \right)}a_{m - n}}}} \\ {\begin{matrix} {0 \leq j < {1{\quad\quad}{and}\quad{with}\quad h_{F\quad 0}}} \\ \left( {{nlT}_{e} + {jT}_{e,{\underset{\_}{T}q\quad\max}}} \right) \end{matrix} = \begin{bmatrix} {h_{F\quad 0}\left( {{nlT}_{e} + {jT}_{e} - \tau_{{q\quad\max},1}} \right)} \\ \vdots \\ {h_{F\quad 0}\left( {{nlT}_{e} + {jT}_{e} - \tau_{{q\quad\max},P_{q\quad\max}}} \right)} \end{bmatrix}} \end{matrix} & (28) \end{matrix}$ for

it is possible to constitute the following observation vector from equation (27): $\begin{matrix} \begin{matrix} {{z\left( {mlT}_{e} \right)} = \begin{bmatrix} {{\hat{s}}_{q\quad\max}\left( {mlT}_{e} \right)} \\ {{\hat{s}}_{q\quad\max}\left( {{mlT}_{e} + T_{e}} \right)} \\ \vdots \\ {{\hat{s}}_{q\quad\max}\left( {{mlT}_{e} + {\left( {I - 1} \right)T_{e}}} \right)} \end{bmatrix}} \\ {= {\sum\limits_{n = {- L_{o}}}^{L_{0}}{{h_{z}(n)}a_{m - n}}}} \\ {{{where}\quad{h_{z}(n)}} = \begin{bmatrix} h_{n,0} \\ h_{n,1} \\ \vdots \\ h_{n,{l - 1}} \end{bmatrix}} \end{matrix} & (29) \end{matrix}$

where h_(n,j)=V_(qmax)h_(F0)(nIT_(e)+jT_(e), τ _(qmax)). From the model of equation (29), it is sufficient to apply an ICA method to the observation z(mIT_(e)) in order to estimate the 2L₀+1 symbol trains {a_(m-n)} such that −L₀≦n≦L₀.

To summarize, this variant comprises the following steps:

Step II.b.1: Determination of the symbol period T by applying a cyclic detection algorithm as in [1] and [10].

Step II.b.2: Sampling of the observations x(t) with I samples per symbol such that T=IT_(e).

Step II.b.3: Application of an ICA method to the observations x(t) in order to obtain ŝ(t) and Â from equation (24).

Step II.b.4: Sorting of the outputs according to Q groups of correlated multipaths in order to obtain Â_(q) and ŝ_(q)(t) for (1≦q≦Q): to do this, a correlation test for all the output pairs i and j with the two-hypothesis test of equation (26). Firstly the outputs associated with the 1st output will be sought by starting the test for 2<j≦P_(Q)×Q and i=1. Next, removed from the list of outputs are all those associated with the 1st that will constitute the 1st group with q=1. The same series of tests is repeated with the other outputs that are not correlated with the 1st output in order to constitute the 2nd group. This operation is continued to the last group where in the end no output channel will remain.

Step II.b.5: Determination of the better group of multipaths where (Â_(q) ^(H)Âq)is a maximum in q=qmax.

Step II.b.6: Constitution of the observation vector z(t) of equation (29) from the signal ŝ_(qmax)(t).

Step II.b.7: Application of an ICA method for estimating the symbol trains {a_(m-n)} where −L₀≦n≦L₀. From the symbol trains is chosen that one which is associated with the higher-modulus vector h_(z)(i), namely {â_(m-i)}.

Step II.b.8: Determination of the phase α_(imax) of the output associated with the higher-modulus vector h_(z)(i) applying steps B.1, B.2 and B.3.

Step II.b.9: Phase-resetting of the symbol trains {â_(m)} by taking {circumflex over (â)}_(m)=â_(m)exp(−jα_(imax)). The symbol train {{circumflex over (â)}_(m)} constitutes the output of the demodulator of this subprocess.

Step II.b.10: Estimation of the propagation channel parameters, namely the angle θ_(q,p) and the delay τ_(q,p). The angles of incidence θ_(q,p) are determined from the Â_(q) values for (1≦q≦Q) applying the MUSIC [1] algorithm to the matrix Â_(q)Â_(q) ^(H). The matrices A_(q) are deduced from these goniometry values in order to deduce therefrom an estimate of s(t, τ _(q)) taking ŝ(t, τ _(q))=A_(q) ^(#)X_(q)(t). Since the elements of the ŝ(t, τ _(q)) are composed of the signals s(t−τ_(q,p)), the delays τ_(p,q)-T_(1,1) are determined by maximizing the c(τ)=|ŝ_(q,p)(t−τ)ŝ_(1,1)(t)*|² critera where ŝ_(q,p)(t) is the p^(th) component of ŝ(t, τ _(q)).

Another way of Implementing the Process Estimation of the Carrier Frequency and Deduction of the {a_(m)} Symbols.

This technique consists in estimating the carrier frequency f₀ of the transmitter or the complex z₀=exp(j2Πf₀T_(e)) in order thereafter to deduce the symbols {a_(m)} from the symbols {b_(m)}, taking, from equation (3): a _(m) =b _(m) exp(−j2Πf ₀ mIT _(e))=b _(m) z ₀ ^(−mI)   (30)

This step is applied after step I.4 of reordering the symbols and the channel vectors. From equations (3), (17), (7) and (8), the following channel vectors are used: $\begin{matrix} {{{\hat{h}}_{z}(n)} = {{\begin{bmatrix} {z_{0}^{nl}{h\left( {nlT}_{e} \right)}} \\ {z_{0}^{{nl} + 1}{h\left( {{nlT}_{e} + T_{e}} \right)}} \\ \vdots \\ {z_{0}^{{nl} + {({l - 1})}}{h\left( {{nlT}_{e} + {\left( {I - 1} \right)T_{e}}} \right)}} \end{bmatrix}\quad{for}\quad n} \in \Omega}} & (31) \end{matrix}$

where Ω={Ind_([rp−L0, rp+L0])(n)=1 for a p such that 1≦p≦P}

Since it is known that Ω={n₁< . . . <n_(Lc)}, a grand vector b is obtained from the vectors ĥ_(z)(n), such that: $\begin{matrix} {w = \begin{bmatrix} {{\hat{h}}_{z}\left( n_{1} \right)} \\ {{\hat{h}}_{z}\left( n_{2} \right)} \\ \vdots \\ {{\hat{h}}_{z}\left( n_{K} \right)} \end{bmatrix}} & (32) \end{matrix}$

The search for f₀ consists in maximizing the following criterion: $\begin{matrix} {{{{Carrier}\quad\left( f_{0} \right)} = {{w^{H}{c\left( {\exp\left( {{j2\pi}\quad f_{0}T_{e}} \right)} \right)}}}^{2}}{{{where}\quad c\left( z_{0} \right)} = {\begin{bmatrix} {c\left( {n_{1},z_{0}} \right)} \\ {c\left( {n_{2},z_{0}} \right)} \\ \vdots \\ {c\left( {n_{K},z_{0}} \right)} \end{bmatrix}\quad{and}}}\text{}{{{where}\quad{c\left( {n,z_{0}} \right)}} = \begin{bmatrix} z_{0}^{nl} \\ z_{0}^{{nl}\quad + \quad 1} \\ \vdots \\ z_{0}^{{nl}\quad + \quad{({l\quad + \quad 1})}} \end{bmatrix}}} & (33) \end{matrix}$

The steps of the process suitable for the case of a transmitter with a non-zero frequency are the following:

Step III.a.1: Step I.1 to step I.4 described above in order to obtain the symbol trains {{circumflex over (b)}_(m-k,)} associated with the channel vectors ĥ_(z)(k_(j)).

Step III.a.2: Construction of the vector w of equation (32) from the ĥ_(z)(k_(j)).

Step III.a.3: Maximization of the carrier (f₀) criterion of equation (33) in order to obtain f₀.

Step III.a.4: Application of equation (30) in order to deduce the symbols {a_(m)} from the symbols {b_(m)}.

Step III.a.5: Step I.5 to step I.7 described above.

In the case of a transmitter with non-zero frequency and for decorrelated multipaths, the steps are the following:

Step III.b.1: Step II.a.1 to Step II.a.4 described above in order to obtain the vector z(t) of equation (22).

Step III.b.2: Application of the ICA methods [4], [5], [15] and [19] in order to estimate L_(c) symbol trains {{circumflex over (b)}_(m,j)} associated with the channel vectors ĥ_(z,j).

Step III.b.3: Reordering of the symbol trains {{circumflex over (b)}_(m,j)} and of the channel vectors ĥ_(z,j) applying steps A.1, A.2 and A.3 in order to obtain the symbol trains {{circumflex over (b)}_(m-k,)} associated with the channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)).

Step III.b.4: Construction of the vector w of equation (32) from the ĥ_(z)(k_(j)).

Step III.b.5: Maximisation of the carrier (f₀) criterion of equation (33) in order to obtain f₀.

Step III.b.6: Application of equation (30) in order to deduce the symbols {a_(m)} from the symbols {b_(m)}.

Step III.b.7: Choice of the symbol train associated with the higher-modulus vector h_(z)(i), namely, {â_(m-i)}.

Step III.b.8: Step II.a.7 to step II.a.9 described above.

In the case of a transmitter with non-zero frequency and for correlated multipaths, the steps are for example the following:

Step III.c.1: Step II.b.1 to step II.b.6 No. 2.2 in order to obtain the vector z(t) of equation (29).

Step III.c.2: Application of ICA methods [4], [5], [15] and [19] in order to estimate the L_(c) symbol trains {{circumflex over (b)}_(m,j)} associated with the channel vectors ĥ_(z,j).

Step III.c.3: Reordering of the symbol trains {{circumflex over (b)}_(m,j)} and of the channel vectors ĥ_(z,j) applying steps A.1, A.2 and A.3 so as to obtain the symbol trains {{circumflex over (b)}_(m-k) _(j) } associated with the channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)).

Step III.c.4: Construction of the vector w of equation (32) from the ĥ_(z)(k_(j)).

Step III.c.5: Maximization of the carrier criterion (f₀) of equation (33) in order to obtain f₀.

Step III.c.6: Application of equation (30) for deducing the symbols {a_(m)} from the symbols {b_(m)}.

Step III.c.7: Choice among the symbol trains of that one which is associated with the higher-modulus vector h_(z)(i), namely {â_(m-i)}.

Step III.c.8 : Step II.b.8 to step II.b.10 described above.

REFERENCES

[1] R. O. Schmidt, “A signal subspace approach to multiple emitter location and spectral estimation”, November 1981.

[2] W. A. BROWN, “Computationally efficient algorithms for cyclic spectral analysers”, 4^(th) ASSP Workshop on Spectrum Modelling, August 1988.

[3] S. V. SCHELL and W. GARDNER, “Detection of the number of cyclostationary signals in unknowns interference and noise”, Proc, Asilonan Conference on Signal, Systems and Computers, 5-9 November 1990.

[4] J. F. CARDOSO and A. SOULOUMIAC, “Blind beamforming for non-Gaussian signals”, IEE Proceedings-F, Vol. 140, No. 6, pp. 362-370, December 1993.

[5] P. COMON, “Independent Component Analysis, a new concept?”, Signal Processing, Elsevier, April 1994, Vol 36, No. 3, pp. 287-314.

[6] S. MAYRARGUE, “A blind spatio-temporal equalizer for a radio-mobile channel using the Constant Modulus Algorithm CMA”, ICASSP 94, 1994 IEEE International- Conference on Acoustics Speech and Signal Processing, 19-22 April 1994, Adelaide, South Australia, pp. 317-319.

[7] E. MOULINES, P. DUHAMEL , J. F. CARDOSO and S. MAYRARGUE, “Subspace methods for the blind identification of multichannel FIR filters”, IEEE Transactions On Signal Processing, Vol. 43, No. 2, pp. 516-525, February 1995.

[8] V. VANDERVEEN, “Joint Angle and delay Estimation (JADE) for signal in multipath environments”, 30^(th) ASILOMAR Conference in Pacific Grove, IEEE Computer Society, Los Alamitos, Calif., USA, 3-6 November 1996.

[9] P. CHEVALIER, V. CAPDEVIELLE, and P. COMON, “Behavior of HO blind source separation methods in the presence of cyclostationary correlated multipaths”, IEEE SP Workshop on HOS, Alberta (Canada), July 1997.

[10] A. FERREOL, patent Ser. No. 98/00731. “Procédé de détection cyclique en diversité de polarisation” [Cyclic detection method in polarization diversity], 23 Jan. 1998.

[11] C. B. PAPADIAS and D. T. M. SLOCK, “Fractionally spaced equalization of linear polyphase channels and related blind techniques based on multichannel linear prediction”, IEEE Transactions On Signal Processing”, March 1999, Vol. 47, No. 3, pp 641-654.

(12] E. DE CARVALHO and D. T. M. SLOCK, “A fast Gaussian maximum-likelihood method for blind multichannel estimation”, SPAWC 99, Signal Processing Advances in Wireless Communications, 9-12 May 1999, Annapolis, US, pp. 279-282.

[13] H. ZENG et L. TONG, “Blind channel estimation using the second-order statistics: Algorithms”, IEEE Transactions On Signal Processing, August 1999, Vol. 45, No. 8, pp. 1919-1930.

[14] A. FERREOL and P. CHEVALIER, “On the behavior of current second- and higher-order blind source separation methods for cyclostationary sources”, IEEE Trans. Sig. Proc., Vol. 48, No.6, pp. 1712-1725, June 2000.

[15] P. COMON, “From source separation to blind equalization, contrast-based approaches”, ICISP 01, Int. Conf. on Image and Signal Processing, 3-5 May 2001, Agadir, Morocco, pp. 20-32.

[16] L. PERROS-MEILHAC, E. MOULINES, K. ABED-MERAIM, P.

CHEVALIER and P. DUHAMEL, “Blind identification of multipath channels: A parametric subspace approach”, IEEE Transactions On Signal Processing, Vol. 49, No. 7, pp. 1468-1480, July 2001.

[17] I. JANG and S. CHOI, “Why blind source separation for blind equalization of multiple channels?”, SAM 02, Second IEEE Sensor Array and Multichannel Signal Processing Workshop, 4-6 August 2002, Rosslyn, US, pp. 269-272.

[18] A. FERREOL, L. ALBERA and P. CHEVALIER, Higher-order blind separation of non zero-mean cyclostationary sources”, (EUSIPCO 2002), Toulouse, 3-6 September 2002, pp. 103-106.

[19] L. ALBERA, A. FERREOL, P. CHEVALIER and P. COMON, “ICAR, un algorithme d'ICA à convergence rapide, robuste au bruit [ICAR, a noise-robust rapidly convergent ICA algorithm”], GRETSI, Paris, 2003.

[20] Z. DING. and J. LIANG, “A cumulant matrix subspace algorithm for blind single FIR channel identification”, IEEE Transactions On Signal Processing, Vol. 49, No. 2, pp. 325-333, February 2001. 

1. A process for the blind demodulation of a linear-waveform source or transmitter in a system comprising one or more sources and an array of sensors and a propagation channel, said process wherein the symbol period T is determined and samples are taken at T_(e) such that T=IT_(e) a spatio-temporal observation z(t), the mixed sources of which are symbol trains from the transmitter, is constructed from the observations x(kT_(e)); an ICA-type method is applied to the observation vector z(t) in order to estimate the L_(c) symbol trains {a_(m-i)} that are associated with the channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)); the L_(c) outputs (â_(m,j), ĥ_(z,j)) are arranged in the same order as the inputs (a_(m,i),h_(z)(i)) so as to obtain the propagation channel vectors ĥ_(z,j)=ĥ_(z)(k_(j)); and the phase α_(imax) associated with the outputs is determined.
 2. The process as claimed in claim 1, wherein the propagation channel parameters are estimated in order to determine the carrier frequency so as to compensate for the symbol trains in order to obtain them in baseband.
 3. The process as claimed in claim 1, wherein it includes a step of estimating the angle θ_(p) and delay τ_(p) parameters of the propagation channel. 